Methods of Voting
a) Majority
b) Plurality
c) Borda Count
d) Hare System
e) Sequential Pairwise
f) Approval Voting
a1)How do you determine a winner in a two person election?
b1) How do you determine a winner in an election with more than two alternatives?
b2) What are some problems/paradoxes with voting different voting methods?
) Determining a winner in a two person election by majority rules.
b-f) Determining a winner in an election with more than two alternatives.
A six question free-response test on voting
Unit 2
Weighted Voting
a) Notation of Weighted Voting
b) Banzahf Power Index
c) Shapley-Shubik Power Index
d) Equivalent Voting Systems
a1) What is the difference between voting and weighted voting?
a2) What is the power of a voter?
b1) What is a critical voter?
b2) How do I determine the Banzahf power of a voter?
c1) What is a pivotal voter?
c2) How do I determine the S-S power of a voter?
d1) How are power indicies calculated for systems when weights are not given?
b1) To be able to calculate the Banzahf power index for any weighted voting system.
c1) To be able to calculate the S-S power index for any weighted voting system.
d1) To be able to calculate the power index for any voting system when the voters' weights are not given.
A 4 question free response test.
Unit 3
Fair Division Procedures
a) Divorce Procedures
b) Inheritance Procedures
c) Cake Cutting Procedures
a1) How do two people divide an estate that is made up of indivisible items?
b1) How do more than two people divide an estate that is made up of indivisible items?
c1) How do two people divide an indivisible item such as a cake or a piece of property when they place different values on the item?
c2) How do more than two people divide an indivisible item such as a cake or a piece of property when they place different values on the item?
a1) Dividing an estate using the Adjusted Winner Divorce Procedure.
b1)Dividing an estate using the Sealed Bid method.
b2) Dividing an estate using the Knaster Inheritance Procedure.
c1) Dividing a cake using the Divide and Choose method.
c2) Dividing a cake using the Lone Divider method.
c3) Dividing a cake using the Last Diminisher method.
c4) Dividing a cake using the Selfridge-Conway method
c4) Dividing an estate using the Trimming to Indivisible Goods method.
An interactive project.
Unit 4
Apportionment
a) Hamilton Method
b) Jefferson Method
c) Webster Method
d) Hill-Huntington Method
a-d1) How does the US decide how many seats each state should get in the House of Representatives?
a-d2) Are the methods used fair?
a-d1) Calculating a standard divisor.
a-d2) Calculating a standard quota.
a-d3) Calculating a rounded quota.
a-d4) Determining a state's final apportionment.
An online project that ends in a written account in a newspaper format.
Unit 5
Game Theory
a) 2-person zero-sum games.
b) Game Trees.
c) 2-person non-zero-sum games.
a1) What strategy should be used in a zero sum game when both players decide their strategy at the same time?
b1) What strategy should be used in a zero sum game when the players decide their strategy at different times?
c1) What strategy should be used in a non-zero sum game when both players decide their strategy at the same time?
a1) Determining saddle points to find pure strategies.
a2) Calculating oddiments to find mixed strategies.
a3) Using graphical methods or domination to reduce the size of the game.
b1) Calculating a game matrix from a game tree.
c1) Finding Nash equilibrium points.
c2) Determining if the Nash points are Pareto optimal.
c3) Finding the solution (if one) of a non-zero sum game.
A four question free response test
Unit 6
Linear Programming
a) Two variable LP.
b) Multi-variable LP.
a1) How do you optimize (maximize or minimize) a two variable linear programming problem?
b1) How do you optimize (maximize or minimize) a multi-variable linear programming problem?
a1) Graphing inequalities.
a2) Find points of intersection of lines.
a3) Solving systems of equations by substitution.
a4) Solving systems of equations by linear combinations.
a5) Solving systems of equations by Gaussian methods.
a6) Solving systems of equations by matrix multiplication.
a7) Solving two variable LP problems using the graphical method.
b1) Solving Standard Maximum multi-variable LP problems using the Simplex method.
b2) Solving Standard Minimum multi-variable LP problems by using Duality.
b3) Solving Non-Standard multi-variable LP problems using the Simplex method.
A five question free response test.
Unit 7
Cryptography
a) Shift ciphers.
b) Matrix ciphers.
c) Public Key ciphers.
d) Code breaking.
a-d1) Why use codes to write messages?
a-c1) How are messages coded and decoded?
d1) How are secret codes broken?
a1) Able to code and decode messages using a shift cipher.
b1) Able to code and decode messages using a matrix cipher.
c1) Able to code and decode messages using a public key cipher.
d1) Able to break any secret codes.
A project that consists of writing a letter to a teacher in code, then decoding the letter of another student.
Unit 8
Graph Theory
a) Euler Paths.
b) Euler Circuits.
c) Eulerizing Graphs.
d) Creating Graphs from models.
e) Hamilton Circuits.
f) Spanning Trees.
g) Steiner Points.
a1) What is the best way to traverse a graph, going across every edge, starting at one point and ending at another?
b1) What is the best way to traverse a graph, going across every edge, starting at one point and ending at that same point?
c1) How can a graph without an Euler circuit be changed so that it would have an Euler circuit?
e1) What is the best way to traverse a graph, going to every vertex, starting at one point and ending at that same point?
f1) What is the best way to connect the verticies of a graph without creating any circuits?
g1) What is the best way to connect verticies?
a-b1) Finding an Euler path or circuit using Fleury's algorithm.
c-d1) Taking a map or model and creating a graph, then adding edges to find an Euler circuit.
e1) Using the Sorted Edges algorithm to find a Hamilton circuit.
e2) Using the Nearest Neighbor algorithm to find a Hamilton circuit.
f1) Being able to identify and find Spanning Trees.
g1) Able to calculate the edges that make up a Steiner Point for triangles, squares and rectangles.
g2) Able to construct a Steiner Point using a compass and straight edge for a triangle.
Table of Contents
Unit 1
a) Majority
b) Plurality
c) Borda Count
d) Hare System
e) Sequential Pairwise
f) Approval Voting
b1) How do you determine a winner in an election with more than two alternatives?
b2) What are some problems/paradoxes with voting different voting methods?
b-f) Determining a winner in an election with more than two alternatives.
Unit 2
a) Notation of Weighted Voting
b) Banzahf Power Index
c) Shapley-Shubik Power Index
d) Equivalent Voting Systems
a2) What is the power of a voter?
b1) What is a critical voter?
b2) How do I determine the Banzahf power of a voter?
c1) What is a pivotal voter?
c2) How do I determine the S-S power of a voter?
d1) How are power indicies calculated for systems when weights are not given?
c1) To be able to calculate the S-S power index for any weighted voting system.
d1) To be able to calculate the power index for any voting system when the voters' weights are not given.
Unit 3
a) Divorce Procedures
b) Inheritance Procedures
c) Cake Cutting Procedures
b1) How do more than two people divide an estate that is made up of indivisible items?
c1) How do two people divide an indivisible item such as a cake or a piece of property when they place different values on the item?
c2) How do more than two people divide an indivisible item such as a cake or a piece of property when they place different values on the item?
b1)Dividing an estate using the Sealed Bid method.
b2) Dividing an estate using the Knaster Inheritance Procedure.
c1) Dividing a cake using the Divide and Choose method.
c2) Dividing a cake using the Lone Divider method.
c3) Dividing a cake using the Last Diminisher method.
c4) Dividing a cake using the Selfridge-Conway method
c4) Dividing an estate using the Trimming to Indivisible Goods method.
Unit 4
a) Hamilton Method
b) Jefferson Method
c) Webster Method
d) Hill-Huntington Method
a-d2) Are the methods used fair?
a-d2) Calculating a standard quota.
a-d3) Calculating a rounded quota.
a-d4) Determining a state's final apportionment.
Unit 5
a) 2-person zero-sum games.
b) Game Trees.
c) 2-person non-zero-sum games.
b1) What strategy should be used in a zero sum game when the players decide their strategy at different times?
c1) What strategy should be used in a non-zero sum game when both players decide their strategy at the same time?
a2) Calculating oddiments to find mixed strategies.
a3) Using graphical methods or domination to reduce the size of the game.
b1) Calculating a game matrix from a game tree.
c1) Finding Nash equilibrium points.
c2) Determining if the Nash points are Pareto optimal.
c3) Finding the solution (if one) of a non-zero sum game.
Unit 6
a) Two variable LP.
b) Multi-variable LP.
b1) How do you optimize (maximize or minimize) a multi-variable linear programming problem?
a2) Find points of intersection of lines.
a3) Solving systems of equations by substitution.
a4) Solving systems of equations by linear combinations.
a5) Solving systems of equations by Gaussian methods.
a6) Solving systems of equations by matrix multiplication.
a7) Solving two variable LP problems using the graphical method.
b1) Solving Standard Maximum multi-variable LP problems using the Simplex method.
b2) Solving Standard Minimum multi-variable LP problems by using Duality.
b3) Solving Non-Standard multi-variable LP problems using the Simplex method.
Unit 7
a) Shift ciphers.
b) Matrix ciphers.
c) Public Key ciphers.
d) Code breaking.
a-c1) How are messages coded and decoded?
d1) How are secret codes broken?
b1) Able to code and decode messages using a matrix cipher.
c1) Able to code and decode messages using a public key cipher.
d1) Able to break any secret codes.
Unit 8
a) Euler Paths.
b) Euler Circuits.
c) Eulerizing Graphs.
d) Creating Graphs from models.
e) Hamilton Circuits.
f) Spanning Trees.
g) Steiner Points.
b1) What is the best way to traverse a graph, going across every edge, starting at one point and ending at that same point?
c1) How can a graph without an Euler circuit be changed so that it would have an Euler circuit?
e1) What is the best way to traverse a graph, going to every vertex, starting at one point and ending at that same point?
f1) What is the best way to connect the verticies of a graph without creating any circuits?
g1) What is the best way to connect verticies?
c-d1) Taking a map or model and creating a graph, then adding edges to find an Euler circuit.
e1) Using the Sorted Edges algorithm to find a Hamilton circuit.
e2) Using the Nearest Neighbor algorithm to find a Hamilton circuit.
f1) Being able to identify and find Spanning Trees.
g1) Able to calculate the edges that make up a Steiner Point for triangles, squares and rectangles.
g2) Able to construct a Steiner Point using a compass and straight edge for a triangle.